Advances in language modeling have paved the way for novel human-AI co-writing experiences. This paper explores how varying levels of scaffolding from large language models (LLMs) shape the co-writing process. Employing a within-subjects field experiment with a Latin square design, we asked participants (N=131) to respond to argumentative writing prompts under three randomly sequenced conditions: no AI assistance (control), next-sentence suggestions (low scaffolding), and next-paragraph suggestions (high scaffolding). Our findings reveal a U-shaped impact of scaffolding on writing quality and productivity (words/time). While low scaffolding did not significantly improve writing quality or productivity, high scaffolding led to significant improvements, especially benefiting non-regular writers and less tech-savvy users. No significant cognitive burden was observed while using the scaffolded writing tools, but a moderate decrease in text ownership and satisfaction was noted. Our results have broad implications for the design of AI-powered writing tools, including the need for personalized scaffolding mechanisms.